Finite Group Databasebeta

← #28486386

Subgroups — #28486386

Order 15104.

554 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433125C211yesmaximal, characteristic
2#21429049C211yesmaximal, characteristic
3#21422812C212yesmaximal, complemented
4#21421951C212yesmaximal, complemented
5C59×#1466C211yesmaximal, characteristic
6C2×#14273727C2211yescharacteristic
7C2×#14273706—21no—
8C2×#14273706—21no—
9#16837474—24no—
10#16837459C2212yes—
11#16837366—21no—
12C2×C22⋊(C59⋊C8)—21no—
13#16837338—21no—
14#16837338—21no—
15#16837305C412yes—
16#16837271C412yes—
17C4×C4⋊Dic59C2211yescharacteristic
18#16837147—24no—
19C2×C4×D4×C59C2211yescharacteristic
20C59×#432C2212yes—
21C59×#323—21no—
22C59×#323—21no—
23C22×Dic118—21no—
24C2×#11745664—22no—
25C22×C59⋊C8—22no—
26#14273727—22no—
27#14273727—22no—
28#14273706—44no—
29#14273706—44no—
30C2×C4⋊Dic59—21no—
31C2×C4⋊Dic59—21no—
32C2×C4⋊Dic59C2×C412yes—
33C2×Dic59⋊C4—22no—
34C2×Dic59⋊C4—22no—
35C2×C22⋊Dic59—22no—
36C2×C22⋊Dic59—22no—
37C2×C4×Dic59C2×C412yes—
38C4⋊(C59⋊C8)D414yes—
39C22⋊(C59⋊C8)—22no—
40C22⋊(C59⋊C8)—22no—
41C22⋊(C59⋊C8)—22no—
42#14273638—22no—
43C59⋊#95—21no—
44C22×D4×C59C2×C411yescharacteristic
45C23×C4×C59D411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C4×D4×C59—22no—
48C4×D4×C59—22no—
49C2×C4⋊C4×C59C2×C411yescharacteristic
50C2×C22⋊C4×C59D411yescharacteristic
51C2×C22⋊C4×C59D411yescharacteristic
52C2×C42×C59C2311yescharacteristic
53C4⋊C8×C59—22no—
54C22⋊C8×C59—22no—
55#97×C59—22no—
56C24×C59C22⋊C411yescharacteristic
57C24×C59C22⋊C411yescharacteristic
58C2×D4×C59—22no—
59C2×D4×C59—22no—
60C2×D4×C59—41no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—42no—
66C22×C4×C59—42no—
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22×C411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×Dic59—22no—
73C22×Dic59—22no—
74C2×Dic118—42no—
75C2×Dic118—42no—
76C2×Dic118—41no—
77C2×Dic118—41no—
78C2×C8×C59—42no—
79C4⋊C4×C59—22no—
80C22⋊C4×C59—22no—
81C22⋊C4×C59—22no—
82C42×C59C2×D412yes—
83C42×C59#4112yes—
84C22⋊Dic59—44no—
85C22⋊Dic59—44no—
86C4⋊Dic59—24no—
87C4⋊Dic59—24no—
88C4⋊Dic59—42no—
89Dic59⋊C4—44no—
90Dic59⋊C4—44no—
91C4×Dic59—42no—
92#11745664—82no—
93C2×C59⋊C8—24no—
94C2×C59⋊C8—82no—
95C23×C59—21no—
96C23×C59—21no—
97C23×C59—21no—
98C23×C59—21no—
99C23×C59—21no—
100C23×C59—41no—
101C23×C59—41no—
102C23×C59—41no—
103C23×C59—41no—
104C23×C59#10112yes—
105C23×C59C2×C22⋊C411yescharacteristic
106D4×C59—44no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—22no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—41no—
113C2×C4×C59—42no—
114C2×C4×C59—42no—
115C2×C4×C59—41no—
116C2×C4×C59—41no—
117C2×C4×C59—81no—
118C2×C4×C59—81no—
119C2×C4×C59C4×D412yes—
120C2×C4×C59#12011yescharacteristic
121C2×C4×C59#12112yes—
122C2×C4×C59#12311yescharacteristic
123C2×Dic59—24no—
124C2×Dic59—42no—
125C2×Dic59—42no—
126C2×Dic59—42no—
127C2×Dic59—42no—
128C2×Dic59—42no—
129Dic118—82no—
130Dic118—82no—
131C8×C59—82no—
132C59⋊C8—44no—
133#9455—591nomaximal, Sylow
134C22×C59—21no—
135C22×C59—21no—
136C22×C59—22no—
137C22×C59—42no—
138C22×C59—42no—
139C22×C59—41no—
140C22×C59—41no—
141C22×C59—41no—
142C22×C59—41no—
143C22×C59—41no—
144C22×C59—81no—
145C22×C59—81no—
146C22×C59#38611yescharacteristic
147C22×C59C2×#10111yescharacteristic
148C22×C59#41311yescharacteristic
149C4×C59—21no—
150C4×C59—22no—
151C4×C59—22no—
152C4×C59—41no—
153C4×C59—41no—
154C4×C59#45312yes—
155Dic59—44no—
156Dic59—82no—
157#2973—591no—
158#2542—591no—
159#1556—592no—
160#1466D5911yescharacteristic
161#1285—592no—
162C2×C59—21no—
163C2×C59—21no—
164C2×C59—42no—
165C2×C59—81no—
166C2×C59#188711yescharacteristic
167C2×C59#190111yescharacteristic
168C2×C59#190911yescharacteristic
169C2×C4⋊Q8—591no—
170C2×C22⋊Q8—1181no—
171C2×C22⋊Q8—1181no—
172C2×C4×D4C2×D5911yescharacteristic
173#435—1184no—
174#432—592no—
175#432C2×D5912yes—
176#407—1181no—
177C2×C22⋊C8—1181no—
178#399—1181no—
179#399—1181no—
180#390—592no—
181#384—592no—
182C4×C4⋊C4—591no—
183#332—1184no—
184#323—21no—
185#323—21no—
186C59#945511yescharacteristic, complemented, Sylow
187C22×Q8—1181no—
188C22×D4C4×D5911yescharacteristic
189C23×C4D23611yescharacteristic
190C23×C4D23611yescharacteristic
191C2×M16—1182no—
192C22×C8—1182no—
193C4⋊Q8—1182no—
194C4⋊Q8—1182no—
195C22⋊Q8—2364no—
196C22⋊Q8—2364no—
197C4×D4—22no—
198C4×D4—22no—
199C2×C4⋊C4—592no—
200C2×C4⋊C4—1181no—
201C2×C4⋊C4—1182no—
202C2×C4⋊C4—1182no—
203C2×C4⋊C4—1181no—
204C2×C4⋊C4C4×D5911yescharacteristic
205C2×C22⋊C4—1182no—
206C2×C22⋊C4—1182no—
207C2×C22⋊C4C22⋊D5911yescharacteristic
208C2×C22⋊C4C22⋊D5911yescharacteristic
209C2×C42—592no—
210C2×C42C22×D5911yescharacteristic
211C4⋊C8—22no—
212C4⋊C8—594no—
213C22⋊C8—22no—
214C22⋊C8—1182no—
215C22⋊C8—1182no—
216C22⋊C8—1182no—
217#97—22no—
218#97—1182no—
219#95—1181no—
220C24(C2×D59)⋊C411yescharacteristic
221C24(C2×D59)⋊C411yescharacteristic
222C2×Q8—2362no—
223C2×Q8—2362no—
224C2×Q8—2361no—
225C2×Q8—2361no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—1182no—
236C22×C4—1182no—
237C22×C4(C2×D59)⋊C411yescharacteristic
238C22×C4(C2×D59)⋊C411yescharacteristic
239C22×C4C2×C4×D5911yescharacteristic
240C22×C4C2×D23611yescharacteristic
241C22×C4C2×C22⋊D5911yescharacteristic
242M16—4722no—
243C2×C8—42no—
244C2×C8—1184no—
245C2×C8—4722no—
246C4⋊C4—22no—
247C4⋊C4—1184no—
248C4⋊C4—1184no—
249C4⋊C4—2364no—
250C4⋊C4—2364no—
251C4⋊C4—2362no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—2364no—
255C22⋊C4—2364no—
256C42—2362no—
257C42D4×D5912yes—
258C42#1174568712yes—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23#1427364512yes—
269C23C2×(C2×D59)⋊C411yescharacteristic
270Q8—4722no—
271Q8—4722no—
272D4—44no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—1184no—
286C2×C4—2362no—
287C2×C4—2362no—
288C2×C4—2362no—
289C2×C4—2362no—
290C2×C4—2362no—
291C2×C4(C22⋊D59)⋊C412yes—
292C2×C4#1427369711yescharacteristic
293C2×C4#1427369912yes—
294C2×C4#1427371111yescharacteristic
295C8—82no—
296C8—2364no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22—81no—
309C22#1683727911yescharacteristic
310C22C2×#1427364511yescharacteristic
311C22#1683738411yescharacteristic
312C4—21no—
313C4—22no—
314C4—22no—
315C4—41no—
316C4—41no—
317C4—4722no—
318C4—2364no—
319C4#1683757212yes—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—81no—
324C2#2142449211yescharacteristic
325C2#2142456211yescharacteristic
326C2#2142461611yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1466C2
socleC22×C59#386
prime radicalC22×D4×C59C2×C4
cosocleC2×C42×C59C23
hypercenter#1466D59
hypocenterC59#9455
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #21422812C2semidirect
4. #21421951C2semidirect
186. C59#9455semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214331252
#214290492
#214228122
#214219512
C59×#14662
#945559